The severe volatility displayed by financial markets in recent years has led many investors to reassess or revise their investment plans and especially their retirement plans. Central bank policies have made the returns from fixed-income investments increasingly unattractive, as interest rates drop to levels last seen in the 1940's. A reasonable outlook for fixed-income investments is that they are unlikely to provide sufficient income in retirement because of low yields, and may in fact lose value if interest rates do eventually rise (since bond market values would then drop). It seems clear that some degree of equity exposure is necessary for investors to have any hope of achieving positive real rates of return, but equity investment entails its own risks.
Additionally, investors and their advisors are becoming aware of the long-run impact of fees, and that this impact is relatively larger in a low-return environment. For instance, assuming 1.5% per year investment fees (these could be a product-level spread in a deferred annuity or CD, or the management expense for an actively managed mutual fund), and ignoring taxes, here are the results of investing $1 for thirty years, at two different gross rates of return:                8% gross return: $10.0627, 6.5% return net of fees $6.61437: therefore fees are $3.44833, and have absorbed 38.05% of the gross return otherwise available;        5% gross return: $4.32194, 3.5% net return net of fees $2.80679: therefore fees are $1.51515, and have absorbed 45.61% of the gross return otherwise available.        
In lower-return environments, a fixed percentage fee (typical of many investment products) absorbs a larger proportion of a lower dollar amount of return. Even apparently moderate fees cause a significant drag on long-term returns, and inflation and taxes exacerbate this effect. Academic research confirms that higher-expense funds do not perform well enough to overcome their expense drag: see for example “Performance and Characteristics of Actively Managed Retail Mutual Funds with Diverse Expense Ratios”, by John A. Haslem, H. Kent Baker, and David M. Smith, available at:
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1155776
Investors have therefore begun to conclude that lower fees are essential, and one sign of this is the surge of interest in exchange-traded funds (ETF's). Although ETF assets are still small relative to conventional mutual fund assets in the United States (ETF assets are still only 8.5% of conventional mutual fund assets, according to the Investment Company Institute: see http://www.ici.org/pdf/2011_factbook.pdf), they are growing much more quickly than conventional mutual fund assets, and the much lower fees charged (as low as 0.06% per year) are undoubtedly part of the reason.
A low-cost diversified portfolio with a sizeable equity component is therefore a key requirement for investors going forward. This is clearly true during the “accumulation phase”: saving for retirement up until say ten years before retirement. However, other considerations (especially guarantees) become important in the “retirement imminent” phase (the decade preceding retirement), as well as once retired.
In the latter two phases, a low-cost diversified portfolio with some level of guarantees is required, as contemplated in e.g. U.S. Pat. No. 7,716,075, U.S. Pat. No. 8,131,612, and
http://aria4advisors.com/wp-content/uploads/2012/01/RetireOne_Transamerica_Investor-Brochure—0112.pdf.
See also:
http://www.sec.gov/Archives/edgar/data/845091/000119312511270213/0001193125-11-270213-index.htm
Guarantees become critical in the retirement phase because capital, once lost, is hard or impossible to replenish in the absence of employment income. Diversification across multiple equity classes is critical to ensure that returns are not subject to the risk of “backing the wrong horse”—for example, the Nikkei index reached its peak in 1989, fell sharply, and has not yet regained its value. See:
http://www.forecast-chart.com/historical-nikkei-225.html
Some degree of low-cost diversification can be achieved through the use of a small number of index funds or ETF's. However, to achieve full diversification a broad variety of assets is required. For example, although an S&P 500 index fund provides diversification amongst 500 American companies making up 70% of the U.S. stock market by market capitalization, the capitalization-weighted nature of the index means that most of the exposure is to large-cap stocks. A truly diversified portfolio would contain many more asset classes than just large-cap U.S. stocks, of course, and might include some or all of the following asset classes:                Bonds (domestic and foreign-currency-denominated)        International large-cap equities        International small-cap equities        International growth equities        International value equities        Cash        Commodities        Convertible bonds        Currency-trading strategies        Emerging market equities        Floating-rate notes        Infrastructure        Junk bonds (domestic and foreign-currency-denominated)        Long-short strategies (e.g. 130/30 funds)        Hedge fund replication strategies        Covered-call strategies        Real-return (inflation-protected) bonds        Real estate        U.S. large-cap equities        U.S. small-cap equities        U.S. growth equities        U.S. value equities        Venture capital investments        Volatility-linked assets        
Another line of reasoning supporting the construction of extremely diverse portfolios is that under reasonable assumptions they provide a real-world approximation of the growth-optimal portfolio (GOP). See for example “Approximating the Growth Optimal Portfolio with a Diversified World Stock Index”, by True Le and Edward Platen, available at:
http://www.qfrc.uts.edu.au/research/research_papers/rp184.pdf
as well as “Capital Growth Theory” by Nils H. Hakansson and William T. Ziemba, available at:
http://www.hakansson.com/nils/papers/capita195.pdf
Le & Platen prove, under reasonable regularity assumptions with respect to asset price processes, that sufficiently diversified long-only portfolios approach the GOP in performance. Historical simulations covering the period 1973-2006 and working with slightly more than 100 world sector indices show results substantially outperforming a market-capitalization-weighted index (MCI)—note that a capitalization-weighted index is normally taken to be the standard benchmark against which to assess portfolio performance.
Fortunately, a more diverse portfolio tends to have lower volatility and hence (all other things being equal) lower hedging costs, so the goals of constructing a diversified portfolio and hedging it to protect capital are broadly compatible in theory. There are, however, computational challenges in calculating the option price sensitivities (“Greeks” further defined below) required for portfolio hedging in the case where the portfolio holds many assets.
An option for which the payoff depends on the weighted average of the performance of multiple assets is usually referred to as a basket option. We can therefore refer to an option where the payoff depends on the weighted average of the performance of a large number of assets as a many-asset basket option.
In order to hedge an option payoff (say for example a minimum accumulation benefit), a financial service provider will usually start by computing the “Greeks” for the payoff: the partial derivatives of the payoff amount with respect to changes in asset value, interest rates, volatility, passage of time, and so on. Common ways to do this involve taking advantage of analytical solutions where they exist, and the use of Monte Carlo simulations using common random numbers and numerical differentiation, where analytic solutions are not available.
The use of Monte Carlo simulation to price options is described in, for example, “Options, Futures, and Other Derivative Securities” by John Hull, Prentice-Hall, © 1989. Numerical differentiation is described in, for example, “Numerical Methods” by Germund Dahlquist and Ake Bjorck, Prentice-Hall, © 1974.
An example of the analytical approach, in the case of an option based on a single asset, is to use the Black-Scholes formula and its partial derivatives to calculate the option value, delta, gamma, rho, vega, and theta. A useful reference for this approach is “Options, Futures, and Other Derivative Securities” by John Hull, Prentice-Hall, © 1989.
An example of the second approach, in the case where a basket option (defined below) payoff based on the performance of many assets must be hedged (in the special case where the assets are mutual funds and are assumed to depend strongly on a small number of underlying indices), is given in U.S. Pat. No. 8,131,612.
In the general case, however, where a many-asset basket option (defined below) is to be hedged, and no special structure based on a small number of underlying indices can be assumed, the Monte Carlo simulation plus finite differencing approach can become very inefficient. For example, to calculate two-sided deltas for payoffs based on a portfolio of 100 assets, a total of 201 simulations using common random numbers must be run: 1 base simulation, 100 simulations in which the starting value for each asset is individually reduced to say 99% of its base value, and 100 simulations in which the starting value for each asset is individually increased to say 101% of its base value.
Clearly, since the computational effort increases linearly within each simulation according to number of assets, and also increases linearly in the number of simulations that must be run in order to calculate deltas, the overall effort goes up quadratically, becoming increasingly burdensome as the number of assets becomes large. For example, if Greek computation takes one second for a 8 asset option, then quadratic growth would imply that Greek computation for a 100 asset option would take about (100/8)2=156.25 seconds. If in contrast linear growth could be achieved, then the computation would take 12.5 seconds.
A key concept in dealing with calculating Greeks for complex payoffs incorporating behavioral assumptions is reverse mode automatic differentiation (also called adjoint differentiation, or backwards differentiation), which is a specific application of the multivariate chain rule of differential calculus, as described in (for example) Theorem 1.8.4 in “Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach” (2nd ed.) by John H. Hubbard and Barbara Burke Hubbard, Prentice Hall © 2002. The matrix of required partial derivatives for application of the multivariate chain rule is known as the Jacobian.
The backwards differentiation concept was first developed in detail by Paul J. Werbos in his 1974 Harvard Ph. D. thesis “Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences”, and is described in some detail (as “the Chain Rule for Ordered Derivatives”) in Section 3.1 of “Backwards Differentiation in AD and Neural Nets: Past Links and New Opportunities” by Paul J. Werbos, available at:
http://www.werbos.com/AD2004.pdf
The concept was later independently rediscovered by B. Speelpenning in his 1980 Ph. D. thesis “Compiling fast partial derivatives of functions given by algorithms”, in which he observed that the number of operations required to compute the partial derivatives of a scalar function with respect to the input variables is bounded above by a fixed constant times the number of operations required to compute the function itself. Depending on what counts as an “operation” this fixed constant is in the range 3 to 5. See “Implementing Automatic Differentiation Efficiently” by David Juedes and Andreas Griewank, Argonne National Laboratory, 1990, available at:
http://www.crpc.rice.edu/softlib/TR_scans/CRPC-TR90057throughTR90088/CRPC-TR90074.PDF
The approach to adjoint differentiation that is often adopted is to explicitly construct Jacobian matrices: see for example:
“Jacobian Code Generated by Source Transformation and Vertex Elimination can be as Efficient as Hand-Coding” by Shaun A. Forth, Mohamed Tadjouddine, John D. Pryce, and John K. Reid, available at: http://www.amorg.co.uk/AD/PUBS/saf_toms04.pdf; and “On the Calculation of Jacobian Matrices by the Markowitz Rule for Vertex Elimination”, by Andreas Griewank and Shawn Reese, 1991, available at: ftp://info.mcs.anl.gov/pub/tech_reports/reports/P267.ps.Z; or “Recipes for Adjoint Code Construction”, by Ralf Giering and Thomas Kaminski, available at: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.7838.
The Jacobian approach is most practical if used with a source code transformation tool, so that the user is not explicitly concerned with sparse matrix operations. There is a large literature on source code transformation, of which the above-cited papers are representative, with the tools ADOL-C and TAMC being well-known.
The source code transformation approach clearly leads to additional complexity and makes debugging more difficult. It is also inapplicable if a language well-suited to the problem domain has no available source code transformation tool (most tools to date focus on Fortran and C).
However, an approach can be developed that does not require the use of sparse matrix operations. See for example “Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming”, by Richard D. Neidinger, available at:
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=SIREAD0000520000030 00545000001&idtype=cvips&doi=10.1137/080743627&prog=normal&bypassSSO=1
In Section 7 of the paper, Neidinger suggests that reverse mode differentiation can be performed via solution of a large sparse set of linear equations, as is contemplated in the previously-cited references, and also describes how, for a small straight-line block of code, computations can be arranged in a directed acyclic graph (“DAG”). Indeed, in compiler theory, it has long been known that computation of variables within a straight-line block of code (a so-called “basic block”) can be represented as a DAG (See Section 12.3, “The DAG Representation of Basic Blocks” in “Principles of Compiler Design”, by Alfred V. Aho and Jeffrey D. Ullman, Addison-Wesley, © 1977, 3rd printing April 1979).
An extension of the DAG representation of basic blocks can be developed that allows programs using backwards differentiation to be written more naturally, i.e. without the use of source code transformation, if a vectorized dynamically-typed language is available.